Question: When a customer places an order at Ying Ying's bakery, there is an $8\%$ probability that the customer will report a food allergy. One day, $12$ customers place orders at Ying Ying's bakery. Assuming that each of the $12$ customers is equally likely to report a food allergy, what is the probability that at least one customer will report a food allergy? Round your answer to the nearest hundredth. $P(\text{at least one with allergy})=$
Solution: Strategy In this situation it is much easier to calculate the probability of the event we are looking for (at least one customer who reports a food allergy) by calculating the probability of its complement (none of the customers report a food allergy), and subtracting from $1$. In other words, we can use this strategy: $P(\text{at least one with allergy})=1-P(\text{none of }12\text{ with allergy})$ Calculations $\begin{aligned} &\phantom{=}P(\text{at least one with allergy}) \\\\ &=1-P(\text{none of }12\text{ with allergy}) \\ \\ &=1-(0.92)^{12} \\ \\ &\approx 1-0.3677 \\ \\ &\approx 0.6323\end{aligned}$ Answer $P(\text{at least one with allergy}) \approx 0.63$